## Math 5285H: Fundamental Structures of Algebra I (Fall 2011)

**Lectures:**MWF 10:10-11:00 in Vincent Hall 20.

**Instructor:**Gregg Musiker (musiker "at" math.umn.edu)

**Office Hours:**MWF 11:00-12:05 in Vincent 251; also by appointment.

## Course Description:

This is the first semester of a course in the basic algebra of groups, ring, fields, and vector spaces. Roughly speaking, the Fall and Spring semesters should divide the topics as follows:Fall) Vector spaces, linear algebra, group theory, symmetry, and the Sylow Theorems

Spring) Rings, fields, and Galois theory

**Prerequisites:**Some previous exposure to linear algebra (vectors, matrices, determinants) (such as from 2243, 2373, or 2573) would help. Also, one should either have the ability to write and read mathematical proofs (such as from 2283, 2574, or 3283), or have the desire and drive to learn how.

**Required text:**

*Algera*, by Michael Artin, (2nd edition 2011, Prentice Hall).

Fall: some (but not all) of Chapters 1-7;

Spring: some (but not all) of Chapters 11,12,13,15,16

Note that I will be following the brand new second edition in this course, and encourage you to do the same. If you have the first edition, your text will more or less contain the same topics (although with occasional exceptions and sometimes in a different order), but the references to section numbers and exercises will not match the ones I announce in class and on the website. If you are in this situation, please find a classmate with the new edition or come talk to me early on.

Title | Author(s) | Year |
---|---|---|

Introduction to abstract algebra | W. Keith Nicholson | 2007 |

Contemporary abstract algebra | Joseph A. Gallian | 1994 |

Algebra | Thomas Hungerford | 1980 |

Abstract Algebra | Dummit and Foote | 2004 |

Abstract algebra: theory and applications | Thomas Judson | 1994 |

## Grading:

**Homework (50%):**There will be 5 homework assignments due approximately every other week (tentatively) on Wednesdays in class. The first homework assignment is due on September 21st.

I encourage collaboration on the homework, as long as each person understands the solutions, writes them up in their own words, and indicates their collaborators. Late homework will not be accepted. Early homework is fine, and can be left in my mailbox in the School of Math mailroom in Vincent Hall 107. Homework solutions should be well-explained; the grader is told not to give credit for an unsupported answer. Complaints about the grading should be brought to me.

**Exams (15% each):**There will be 2 take-home exams, handed out on October 5th (due October 12th) and November 9th (due November 16th). Each will be open book, open library, open notes, and with calculators allowed. However, for these exams, you are not allowed to consult other electronic sources, such as the internet, and you are not allowed to collaborate or consult with other students or other human sources. These exams are to be collected in class.

**Final Exam (20%):**The final exam will be take-home as well, under the same policies as above, handed out on December 7th, to be turned in during class on December 14th.

## Class Participation:

Participation in class is encouraged. Please feel free to stop me and ask questions during lecture. Otherwise, I might stop and ask you questions instead.## Course Syllabus and Tentative Lecture Schedule

## Announcements

Assignment or Exam | Due date | Problems from Artin text, unless otherwise specified |
---|---|---|

Homework 1 | Wednesday 9/21 |
Chapter 1: # 1.4, 1.6, 1.7, 1.10, 1.11, 1.13, 2.2, 2.5, 2.6, 2.10,
3.1, 4.1, 4.3, 5.1, 5.3, 5.4, 6.2, M.3, M.7 |

Homework 2 | Wednesday 10/5 |
Chapter 2: # 1.3, 2.1, 2.2, 2.4, 3.1, 3.2, 4.1, 4.3, 4.6, 4.7, 4.8, 4.9,
5.2, 5.3, 5.6, 6.2, 6.3, 6.6, 6.7, 6.10 |

Exam 1 | Wednesday 10/12 |
Download Here |

Homework 3 | Wednesday 10/26 |
Chapter 3: # 1.2, 1.5, 1.6, 1.8, 1.9, 1.10, 1.11, 3.1,
3.2, 4.3, 4.4, 5.1, 5.2 Chapter 4: # 1.3, 1.4, 2.3 IMPORTANT NOTE: Some books number the exercises differently. In Chapter 3, Problems 1.2, ..., 1.11 should be in the section labeled "Fields", Problems 3.1, 3.2 should be in the section labeled "Bases and Dimension", Problems 4.3, 4.4 should be in the section labeled "Computing with Bases", Problems 5.1, 5.2 should be in the section labeled "Direct Sums". |

Homework 4 | Wednesday 11/9 |
Chapter 4: # 3.2, 4.1, 4.2, 6.8 Chapter 5: # 1.1, 1.3, M.1 (for n=2 and 3), M.5 Typo in M.5: formula should be 1/\alpha ( [(1+\alpha)/2]^n - [(1-\alpha)/2]^n) Chapter 6: (Section 3 Isometries of the Plane): # 3.2, 3.3, 3.4, (Section 4 Finite Groups of Orthogonal Operators on the Plane): # 4.1, 4.2, 4.3, (Section 5 Discrete Groups of Isometries): # 5.1, |

Exam 2 | Wednesday 11/16 |
Download Here |

Homework 5 | Wednesday 12/7 |
Chapter 6: # 7.1, 7.3, 8.2, 10.2, 11.1, 12.4, 12.5 Chapter 7: # 2.1, 2.2, 2.3, 2.7, 2.8, 2.9, 2.14, 2.17, 3.2, 3.3, 3.4, 4.4, 4.5 |

Final Exam | Wednesday 12/14 |
Download Here |